Is It Dead Yet? - the odds of killing vehicles in 40k

Summary: How to calculate the odds of killing a vehicle in a single round of shooting.

How I calculate the effectiveness of anti-vehicle countermeasures.

I tend to throw odds around, and I've had a couple of people ask me how I come up with them. I make computers do the work. I built a spreadsheet using the techniques that GreatBigTree describes over here, and added a couple of twists of my own. Figured I'd make it available for others if they're interested. I've saved it in a fairly generic Excel format, but you ought to be able to open it with LibreOffice or one of the other free downloads. If you're at all interested in how it works, I encourage you to read through the next bit and check my maths; if you just want to plug in some numbers and look at the answers, skip down to the section titled "The Spreadsheet".

The Maths

In this game, the usual way to calculate how well a thing works against another thing is to calculate the average number of wounds that it will inflict. For infantry, that's fair enough, but for vehicles it belies the number I most often care about more: what are the odds that I'll kill it.

There are a couple of basic principles I use to work out these odds:

1) Its often easier to work out the chance of _not_ doing something than it is to work out the odds of doing something. Then the odds of your thing happening is just 1.0 minus the odds of it not happening. This is especially true when you're trying to answer questions like "the odds of 3 or more hull points"; when you take a lot of shots, the "or more" covers a lot of ground, but you don't need to work those out. You just work out the odds of zero, one, or two, and subtract from 1.0. I blatantly stole this technique from a post by GreatBigTree ([6th Ed] How to Generate Statistics - Part 1: Number of Casualties Spread), so for more detail, go check that one out.

2) A simple formula for re-rolls, which is based on the above. With re-rolls you have to take into account that you only re-roll if you don't succeed, which is a trifle fiddly. Far simpler to work out the odds of failing twice, and subtracting from 1.0. So if I have a 1 in 6 chance of succeeding with each roll, and I get a re-roll, then I have a 5/6 chance of failing each roll, and a (5/6)^2 = 25/36 chance of failing twice in a row. Which means I have an 11/36 chance of succeeding on one or more of my rolls (which is all I need to succeed overall if I have a re-roll.) So if S is my odds of success with each roll, (1.0 - S) is my odds of failing with each roll, (1.0 - S)^2 is my odds of failing on both rolls, and 1.0 - (1.0 - S)^2 is my odds of success with a re-roll.

But what if I'm not sure if I'll get a re-roll? If R is the probability I'll get a re-roll, then 1.0 - R is the probability that I won't (one of the two has to happen) and my odds of success overall are:

3) Combining odds. We can add up probabilities so long as they're mutually exclusive. We're working out the odds of a single weapon type killing a vehicle by working out the odds of the vehicle surviving; the odds that we get 0, 1, or 2 hull points without it exploding. Since we've worked out those three values separately for each weapon, we can work out the odds of the vehicle surviving fire from two weapons combined:

Odds of exactly 1 HP with A+B = (Odds of exactly 1 with A) * (Odds of exactly 0 with B)+(Odds of exactly 0 with A) * (Odds of exactly 1 with B)

Odds of exactly 2 HP with A+B = (Odds of exactly 2 with A) * (Odds of exactly 0 with B)+(Odds of exactly 1 with A) * (Odds of exactly 1 with B)+(Odds of exactly 0 with A) * (Odds of exactly 2 with B)

All of the terms are mutually exclusive, so the addition is safe. And because we've worked out all of the same values for the combo, it is possible to combine a third weapon to the mix of the first two in exactly the same way.

Put your own values into the columns with names in bold. You can override the other columns too, but you'll erase my formulas, so its better, for instance, to set your BS to 1 than to manually set % Hit to 1/6 for a Snap Shot.

BS - doesn't handle values larger than 6, because it rarely seems to come up and I couldn't be bothered. Set to 1 if you're taking Snap Shots.

shots - combine all your shots from weapons with a similar profile together. Combining different profiles for mixed arms is trickier, and I'll mention it below.

S - of the weapon.

AP - of the weapon.

AV - of the target.

save - cover / invul of the target. The actual value; 5for a 5+ save, 7 if they get none. Since we're talking about vehicles, I assume AP has no effect on saves. If your weapon ignores cover, set the value accordingly.

open-top - 1 if the vehicle is open-topped, 0 otherwise.

single hit - odds of a hit on a single roll.

re-roll hit % - chance of a re-roll on to-hit. 0 = none. 1 = always (e.g. twin-linked.) If, for instance, someone were to invent a weapon that fired first and gave you twin-linked if it scored at least one hit, you could put the odds of getting at least one hit with that weapon in here.

To Hit - your odds of getting a hit, counting re-rolls.

single damage - odds of doing a HP on a single roll.

re-roll pen % - chance of a re-roll on the Armour Penetration roll. I assume you only re-roll misses - dealing with different strategies for re-rolling glances is annoying, so I didn't.

To Damage - your odds of doing a HP, counting re-rolls.

single pen - odds of a penetrating hit on a single roll.

To Pen - your odds of getting a penetrating hit, counting re-rolls.

To Fail Save - the odds that the target's save will _fail_.

% Hull - odds of hitting, doing a hull point, and bypassing any save, with a single shot.

% Pen - odds of hitting, doing a penetrating hit, and bypassing any save, with a single shot.

single Dest - odds of a Destruction result, per penetrating hit.

% Destroyed - odds of hitting, doing a penetrating hit, bypassing any save, and getting a Destroyed result, with a single shot.

% Hull w/o Destroyed - odds of hitting, doing a hull point, bypassing any save, and failing to get a Destroyed result, with a single shot.

% n+ or Destroyed - odds of doing n or more hull points, with all of your shots, or getting a Destroyed result.

and finally - the point of the whole exercise:

% 3+ or Destroyed - the odds the bloody thing died.

Mixed Arms

There is a combo row in the sheet. Combine all of the shots for all weapons with the same profile into one normal row of the sheet. Do the second weapon profile on the second row of the sheet, without leaving any space between them. Then copy the combo line onto the third row. If there is a third weapon profile, put it on the line immediately following the combo row, and then follow it with another combo row.

An Example

Lets pick a made-up example, so the Warp Lawyer Elite Hit Squad does not descend. I have a vehicle which I shall call a Space-Elf AT-AT; it comes in squadrons of up to 3. It has various weapon choices that can be chosen for each of two different weapon mounts. One weapon I shall call a Splatter Gun; it is S5, Assault 5, AP5, and has a special rule that it will cause other weapons on the vehicle to be twin-linked if it hits. One weapon I shall call Tight Pants (those Space-Elfs wear some wicked Tight Pants.) The sight of these trousers of power causes such searing pain in the viewer that they are S7, Assault 1, AP1. Lets sick a 3 AT-AT squad - each mounting one of each weapon - on a flying angelic waterfowl that, with an AV of 11 and a 5+ invul save, is so hellishly difficult to kill that it is called a HelllDuck. (3 L's, cuz its so ludicrous.) We'll throw in a third profile to demonstrate how that works; Space Elf Warp Ladybugs are a terror to behold at S6 Assault 2 AP-.

Lawyers please note that these do not in any way have the same stats as any actual GW product that I am aware of.

Copy the spreadsheet line and put in the stats for the three gun types. Shooting at a zooming flyer, so BS is 1. Number of shots is for the whole unit (so 3x the weapon). The re-roll hit % of the Tight Pants is 1.0 minus the odds of missing with all 5 shots of a Splatter Gun in a row. Copy the combo line to immediately below the first two, then copy it again after every weapon profile from the third on. Overall, my valiant AT-ATs and friends have slightly more than a 10% chance of taking out the dreaded HelllDuck. They should probably pick another target.